Download §2.1 Probabilities, Events, and Equally Likely Outcomes

Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
§2.1 Probabilities, Events, and
Equally Likely Outcomes
Tom Lewis
Fall Semester
2014
Sample spaces, outcomes and events
The definition of probability
Outline
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
The uniform probability measure
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Definition
An element of a sample space is called an outcome. A subset of a
sample space is called an event.
Problem
Toss a coin three times in a row and record the result after each
toss. Construct the event that...
• exactly 2 heads are tossed;
• exactly 3 heads are tossed;
• at least 2 heads are tossed.
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Problem
4 balls (labelled 1 through 4) are placed in an urn. An experiment
consists of taking two balls from the urn (one at a time and
without replacement). Construct the event that ...
• the sum of the balls is 4;
• the number on the second ball is greater than the number on
the first.
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Definition
A certain experiment has a sample space X of the form
X = {s1 , s2 , . . . , sn }.
To each outcome si we associate a number wi called the weight or
probability of the outcome. The weights must satisfy:
• 0 6 wi 6 1 for all 1 6 i 6 n;
• w1 + w2 + · · · + wn = 1
Here is some common notation: wi = P(si ).
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Definition
Let E ⊂ X be an event. We define the probability of the event E
by the rule
P(E ) = the sum of the weights of the outcomes that comprise E .
Example
Thus, if E = {s1 , s2 , s3 }, we would have P(E ) = w1 + w2 + w3 .
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Problem
Let X = {a, b, c, d , e} with
P(a) = .1, P(b) = .1, P(c) = .2, P(d ) = .3 and P(e) = .3. Find
P(E ) if...
• E = {a, b, c};
• E = {d , e};
• E = {b, c}.
Sample spaces, outcomes and events
The definition of probability
Theorem (The Complement Rule)
Let E ⊂ X be an event. Then
P(E ) + P(E 0 ) = 1.
The uniform probability measure
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Definition
The uniform probability measure is the probability measure that
assigns equal probability to every outcome in the sample space.
Problem
Let X = {a, b, c, d , e, f , g , h, i, j} be the sample space of a
probability experiment.
• How much weight does a uniform probability measure assign
to each outcome?
• Under a uniform measure, what is the probability of the event
E = {b, d , f , g }?
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Uniform probability measures
Let X be a uniform probability space and let E ⊂ X . Then
P(E ) =
n(E )
n(X )
In other words, the probability of an event is the ratio of the
number of outcomes in E to the number of outcomes in X .
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Problem
Toss three coins. What is the probability of tossing at least 2
heads? What is the probability of tossing exactly 2 heads?
Sample spaces, outcomes and events
The definition of probability
The uniform probability measure
Problem
Toss a red and a green die. What is the probability that the dice
match? What is the probability that the dice sum to 7? What is
the probability that the dice sum to 4?